Density operators master equation method

There are several theoretical approaches that can be used to calculate the dynamics and correlation properties of two atoms interacting with the quantized electromagnetic held. One of the methods is the wavefunction approach in which the dynamics are given in terms of the probability amplitudes [9]. Another approach is the Heisenberg equation method, in which equations of motion for the atomic and held operators are found from the Hamiltonian of a given system [10], The most popular approach is the master equation method, in which the equation of motion is found for the density operator of an atomic system weakly coupled to a system regarded as a reservoir [7,8,41], There are many possible realizations of reservoirs. The typical reservoir to which atomic systems are coupled is the quantized three-dimensional multimode vacuum held. The major advantage of the master equation is that it allows us to consider the evolution of the atoms plus held system entirely in terms of atomic operators. 

All that remains to be done for determining the fluctuation spectrum is to compute the conditional average, Eq. (31). However, this involves the full equations of motion of the many-body system and one can at best hope for a suitable approximate method. There are two such methods available. The first method is the Master Equation approach described above. Relying on the fact that the operator Q represents a macroscopic observable quantity, one assumes that on a coarse-grained level it constitutes a Markov process. The microscopic equations are then only required for computing the transition probabilities per unit time, W(q q ), for example by means of Dirac s time-dependent perturbation theory. Subsequently, one has to solve the Master Equation, as described in Section TV, to find both the spectral density of equilibrium fluctuations and the macroscopic phenomenological equation. 

The traditional method to analyze conditions for modification of spontaneous emission is to derive equations of motion for the probability amplitudes or density matrix elements and solve them by direct integration, or by a transformation to easily solvable algebraic equations. Here, we discuss an alternative approach proposed by Akram et al. [24] that allows us to identify conditions for a modification of spontaneous emission directly in the master equation of two arbitrary systems. In this approach, we introduce linear superpositions of the dipole operators... 

The dynamics of the collective two-atom system in a squeezed vacuum can be determined from the master equation of the density operator of the system or from the equations of motion for the transition probability amplitudes [22]. In Section II.B, we derived the master equation for the density operator of a two atom system interacting with the ordinary vacuum field. It is our purpose to extend the master equation to the case of a squeezed vacuum field. The method of derivation of the master equation is a straightforward extension of that presented in Section II.B. 

Various methods have been developed that interpolate between the coherent and incoherent regimes (for reviews see, e.g. (3)-(5)). Well-known approaches use the stochastic Liouville equation, of which the Haken-Strobl-Reineker (3) model is an example, and the generalized master equation (4). A powerful technique, which in principle deals with all aspects of the problem, uses the reduced density matrix of the exciton subsystem, which is obtained by projecting out all degrees of freedom (the bath) from the total statistical operator (6). This reduced density operator obeys a closed non-Markovian (integrodifferential) equation with a memory kernel that includes the effects of (multiple) interactions between the excitons and the bath. In practice, one is often forced to truncate this kernel at the level of two interactions. In the Markov approximation, the resulting description is known as Redfield theory (7). 

There are several ways of proceeding from here to arrive at the master equation for the reduced density matrix. The Zwanzig projection operator technique in Liouville space or the Kubo cumulant expansion may be used and both methods have recently been applied to study optical dephasing in solids. 

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